Locus
of a Moving Point

how to determine the locus of a moving points
that will satisfy the condition.

Locus and Equation to a Locus:

If a point moves on a plane satisfying some given
geometrical condition then the path trace out by the point in the plane is
called its locus. By definition, a locus is determined if some geometrical
condition are given. Evidently, the co-ordinate of all points on the locus will
satisfy the given geometrical condition. The algebraic form of the given
geometrical condition which is satisfy by the co-ordinate of all points on the
locus is called the equation to the locus of the moving point. Thus, the
co-ordinates of all points on the locus satisfy its equation of locus: but the
co-ordinates of a point which does not lie on the locus, do not satisfy the
equation of locus. Conversely, the points whose co-ordinates satisfy the equation
of locus lie on the locus of the moving point.

Examples of Locus
of a Moving Point:

1.
A point moving in such a manner that three times of distance from the
x-axis is grater by 7 than 4 times of its distance form the y-axis; find
the equation of its locus.

Solution: Let P (x, y)
be any position of the moving point on its locus. Then the distance of P from
the x-axis is y and its distance from the y-axis is x.

By problem, 3y – 4x = 7,

Which is the required equation to the
locus of the moving point.

2. Find the equation
to the locus of a moving point which is always equidistant from the points (2,
-1) and (3, 2). What curve does the locus represent?

Solution:

Let A (2, -1) and B (3, 2) be the given
points and (x, y) be the

co-ordinates of a point P on the required locus. Then,

PA2 = (x - 2)2 + (y + 1)2 and PB2 = (x - 3)2 + (y - 2)2

By problem, PA = PB or, PA2 = PB2

or, (x - 2)2 + (y + 1)2 = (x - 3)2 + (y - 2)2

or, x2 - 4x + 4 + y2 + 2y + 1 = x2 – 6x + 9 + y2 – 4y + 4

or, 2x + 6y = 8

or, x + 3y = 4………
(1)

Which is the required equation to the
locus of the moving point.

Clearly, equation (1) is a first degree
equation in x and y; hence, the locus of P is a straight line whose equation is
x + 3y = 4.

3. A and B are two given point
whose co-ordinates are (-5, 3) and (2, 4) respectively. A point P moves in such
a manner that PA : PB = 3 : 2. Find the equation to the locus traced out by P.
what curve does it represent?

Solution: Let (h, k) be the co-ordinates
of any position of the moving point on its locus. By question,

PA/PB = 3/2

or, 3 ∙ PB = 2 ∙ PA

or, 9 ∙ PB2 = 4 ∙ PA2

Or, 9[(h - 2)2 + (k - 4)2] = 4[(h + 5)2 + (k - 3)2]

or, 9 [h2 - 4h + 4 + k2 - 8k + 16] = 4[h2 + 10h + 25 + k2 - 6k + 9]

Or, 5h2 + 5k2 – 76h – 48k + 44 = 0

Therefore , the required equation to the locus traces out by P is

5x2 + 5y2 – 76x – 48y + 44 = 0 ……….. (1)

We see that the equation (1) is a second degree equation in x, y and its coefficients of x2 and y2 are equal and coefficients of xy is zero.

Therefore, equation (1) represents a circle.

Therefore, the locus of P represents the equation of a circle.

4. Find the locus of a moving point
which forms a triangle of area 21 square units with the point (2, -7) and (-4, 3).

Solution: Let the given point be A (2,
-7) and B (-4, 3) and the moving point P (say), which forms a triangle of area
21 square units with A and B, have co-ordinates (x, y). Thus, by question area
of the triangle PAB is 21 square units. Hence, we have,